Monte Carlo study of the planar spin model

We have simulated the two-dimensional classical planar spin model using the Metropolis Monte Carlo technique. The loss of long-range order as a function of the size of the lattice was confirmed. The energy and specific heat were calculated for a square lattice of 900, 3 600, and 10 000 spins. A sharp specific-heat peak was found at $\frac{{k}_{B}T}{J}=1.02$ $J$ is the nearest-neighbor coupling), 15% above the transition temperature $\frac{{k}_{B}{T}_{c}}{J}=0.89$. ${T}_{c}$ was determined by fitting the spin-spin correlation function and the susceptibility to the forms of the Kosterlitz-Thouless theory. The density of vortex pairs was computed and found to increase exponentially with inverse temperature. At ${T}_{c}$ vortex pairs begin to unbind and also larger clusters of vortices appear and unbind as the temperature is increased. These larger clusters may be responsible for the specific-heat peak being sharper and closer to ${T}_{c}$ than simple theories predict.